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Unformatted text preview: 12.5 Homomorphisms – A Generalization of Colourings 663 Lemma 12.5.15 [412] Let H be a core. If the ˜ Hcolouring problem is NP complete, then so is the Hcolouring problem. ut h v j ( c ) ( b ) ( a ) Figure 12.5 Illustrating the subindicator construction; (a) a digraph H with a special vertex h ; (b) the subindicator J with special vertices j,v ; (c) the result ˜ H of applying the subindicator construction with respect to ( J,j,v ) to ( H,h ). To illustrate how to use the indicator and the subindicator construction, let us show that, if H is the digraph in Figure 12.4(a), then the Hcolouring problem is NPcomplete. First apply the indicator construction with respect to the indicator shown in Figure 12.4(b) to H . This gives us the digraph H * in Figure 12.4(c). By Lemma 12.5.14, Hcolouring is NPcomplete if and only if H *colouring is NPcomplete. Now let J be the subindicator consisting of the complete biorientation of a 3cycle with one vertex labelled j and an isolated vertex v 1 . Let H be the result of applying the subindicator construction with respect to ( J,j,v 1 ) to H * . Since v 1 is isolated, a vertex from H * will be in H precisely when it is itself on a complete biorientation of a 3 cycle in H * . Hence H is the complete biorientation of a 3cycle. By Theorem 12.5.10 Hcolouring is NPcomplete and now we conclude by Lemma 12.5.15 that H *colouring and hence also Hcolouring is NPcomplete. Although the subindicator and the indicator constructions are very useful tools for proving the NPcompleteness of many Hcolouring problems, there are digraphs H for which another approach such as a direct reduction from a different type of NPcomplete problem is needed. Such reductions are often from some variant of the satisfiability problem (see Section 1.10). The reader is asked to give such a reduction in Exercise 12.29. For examples of other papers dealing with homomorphisms in digraphs see [135] by Brewster and MacGillivray, [416] by Hell, Zhou and Zhu, [590] by Neˇ setˇ ril and Zhu, [680] by Sophena and [761, 762] by Zhou. 664 12. Additional Topics 12.6 Other Measures of Independence in Digraphs The definition of independence of vertex subsets in digraphs used in this book is by no means the only plausible definition of independence in digraphs. One may weaken the definition of independence in directed graphs in at least two other ways, both of which still generalize independence in undirected graphs. (1) By considering induced subdigraphs which are acyclic. This gives rise to the acyclic independence number , α acyc ( D ), which denotes the size of a maximum set of vertices X such that D h X i is acyclic. (2) By considering induced subdigraphs which contain no 2cycles. This gives rise to the oriented independence number , α or ( D ), which denotes the size of a maximum set of vertices Y such that D h Y i is an oriented graph....
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 Spring '11
 Algorithms

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